Invariants
| Base field: | $\F_{37}$ |
| Dimension: | $1$ |
| L-polynomial: | $1 + 2 x + 37 x^{2}$ |
| Frobenius angles: | $\pm0.552568456711$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-1}) \) |
| Galois group: | $C_2$ |
| Jacobians: | $8$ |
| Isomorphism classes: | 8 |
This isogeny class is simple and geometrically simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
This isogeny class is ordinary.
| $p$-rank: | $1$ |
| Slopes: | $[0, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $40$ | $1440$ | $50440$ | $1872000$ | $69356200$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $40$ | $1440$ | $50440$ | $1872000$ | $69356200$ | $2565781920$ | $94931313160$ | $3512478528000$ | $129961762513960$ | $4808584361239200$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 8 curves (of which 0 are hyperelliptic):
- $y^2=x^3+24 x+11$
- $y^2=x^3+34 x+31$
- $y^2=x^3+20 x+3$
- $y^2=x^3+8 x+8$
- $y^2=x^3+x+2$
- $y^2=x^3+15 x+15$
- $y^2=x^3+3 x$
- $y^2=x^3+33 x+29$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{37}$.
Endomorphism algebra over $\F_{37}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-1}) \). |
Base change
This is a primitive isogeny class.
Twists
Below is a list of all twists of this isogeny class.
| Twist | Extension degree | Common base change |
|---|---|---|
| 1.37.ac | $2$ | (not in LMFDB) |
| 1.37.am | $4$ | (not in LMFDB) |
| 1.37.m | $4$ | (not in LMFDB) |